In the paper Monotonicity for Elliptic Equations in Unbounded Lipschitz Domains - Berestycki - Caffarelli - Niremberg, they use the estimate "obtained by interior elliptic regularity": if $B=B_\rho(y)\subset\Omega$ and $|x-y|\leq\rho/2$, then
$$|u(x)-u(y)|\leq \widehat C\max_Bu\cdot\left(\frac{|x-y|}{\rho}\right)^\gamma+\widehat C\rho^{2-\gamma}|x-y|^\gamma,$$
for $\gamma\in(0,1]$. Someone can help me to obtain this estimate? Thank you.

$\begingroup$The $\gamma$ is used to prove that $u$ is Hölder Continuous. In the paper, we use this estimate to prove this. Can I use this estimate argument for each $\gamma\in(0,1]$? 5PM, you answered an another question to me here: math.stackexchange.com/questions/284506/… but I have one more doubt, about the uniform convergence of the Laplacian (the $C^2$-convergence). Do you know what implies this convergence? Thank you very much!!!$\endgroup$
– José CarlosFeb 4 '13 at 18:35

$\begingroup$I didn't understand why the right hand side is a decreasing function of $\gamma$. Could you explain it?$\endgroup$
– José CarlosFeb 4 '13 at 19:29

$\begingroup$@JoséCarlos I updated my answer with more detail. I'll get back to the convergence question..$\endgroup$
– user53153Feb 4 '13 at 19:52

$\begingroup$Thank you very much @5PM.$\endgroup$
– José CarlosFeb 4 '13 at 19:53